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Identity and Search in Social Networks D.J.Watts, P.S. Dodds, M.E.J. NewmanPowerPoint Presentation

Identity and Search in Social Networks D.J.Watts, P.S. Dodds, M.E.J. Newman

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### Identity and SearchinSocial NetworksD.J.Watts, P.S. Dodds, M.E.J. Newman

### Thank You!Any Questions???

Outlines

- Introduction
- The Hierarchical Model
- Discussion

Milgram’s Experiment

- Short chains of acquaintances exist.
- People are able to findthese chains using only local information.

Results in Literature

- Connected random networks have short average path lengths:
xij log(N)

- N = population size,
- xij = distance between nodes i and j.

Results in Literature

- Kleinberg (2000) demonstrated that emergence of the second phenomenon requires special topological structure.
- For each node i:
- local edges d(i,j) ≤ p
- long-range directed edges
to q random nodes

Pr(ij) ~ d(i,j)-a

Results in Literature

- If networks have a certain fraction of hubs can also search well.
- Basic idea: get to hubs first
- Hubs in social networks are limited.

Hierarchical Model – Why? How?

- Basic idea: impose some high-level structure, and fill in details at random.
- Incorporate identity.
- Need some measure of distance between individuals.
- Some possible knowledge:
- Target's identity, friends' identities, friends' popularity, where the message has been.

Hierarchical Network Construction

- xij = the height of the lowest common ancestor
level between i and j

- z connections for each node with probability:
p(x) = ce-αx

Network constructed from template

Hierarchical template for the network

Hierarchical Network Construction

- Individuals hierarchically partition the social world in more than one way.
- h = 1, …, H hierarchies

- Identity vector
- is position of node i in hierarchy h.

- Social distance:

Directing Messages

- At each step the holder i of the message passes it to one of its friends who is closest to the target t in terms of social distance.
- Individuals know the identity vectors of:
- themselves,
- their friends,
- the target.

Expected Number of Steps

- What is the expected number of steps to forward a message from a random source to a random target?
- Define q as probability of an arbitrary message chain reaching a target.
- Searchable network: Any network for which
q≥ r

for a desired r.

Number of Steps - Results

- If message chains fail at each node with probability p, require
where L = length of message chain.

- Approximation:
L ln r / ln (1 - p)

q = (1 - p)L ≥ r

Searchable Network Regions

- In H-αspace
- p = 0.25, r = 0.05
- b = 2
- g = 100, z = 99
- N=102400
- N=204800
- N=409600

Probability of Message Completion

- α = 0 (squares) versus α= 2 (circles)
- N = 102400
- q ≥ r
q < r

r = 0.05

Milgram's Data

- N = 108
- b = 10
- g = 100
- z = 300
- Lmodel 6.7
- Ldata 6.5
- α = 1, H = 2

Is this an acceptable model?

- Simple greedy algorithm.
- Represents properties present in real social networks:
- Considers local clustering.
- Reflects the notion of locality.

- High-level structure + random links.

Can the Model be Extended?

- Should we consider other parameters such as friend’s popularity information in addition to homophily?
- Allow variation in node degrees?

- Assume correlation between hierarchies?
- Are all hierarchies equally important?

Applications

- Can solutions to sociology problems inform other areas of research?
- Suggested applications:
- Construction of peer-to-peer networks.
- Search in databases.

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